Study the variations of the function $t \mapsto t\ln(t)$ on $\mathbf{R}_+^*$. Verify that we can extend the function by continuity at $0$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $h _ { n }$ the map from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad h _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \cdots + x _ { n } - 1 .$$
We admit that $F _ { n }$ and $h _ { n }$ are both of class $\mathcal { C } ^ { 1 }$ on $U _ { n }$. We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Determine the gradient of $F _ { n }$ and the gradient of $h _ { n }$ at every point of $U _ { n }$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Prove that the restriction of $F _ { n }$ to $\overline { U _ { n } } \cap H _ { n }$ admits a maximum.

We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$. Deduce that for all $x, y \in \mathbb{R}$, denoting $\tilde{x} := x - \tau f'(x)$ and $\tilde{y} := y - \tau f'(y)$, we have $$|\tilde{x} - \tilde{y}|^2 \leq |x-y|^2(1 - \alpha\tau(2 - L\tau)).$$

We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$. We suppose $0 < \tau < 2/L$. Show that $\left|x_n - x_*\right| \leq \rho^n \left|x_0 - x_*\right|$, where $\rho$ is a constant that we will specify, and such that $0 \leq \rho < 1$.

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. Show that $0 \leq f(x) - f(x_*) \leq |x - x_*||f'(x)|$ for all $x \in \mathbb{R}$.

We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := x_n - \tau f'(x_n)$. Show that for all $n \in \mathbb{N}$, assuming $x_0 \neq x_*$, $$f(x_{n+1}) \leq f(x_n) - \frac{\tau}{2}(2 - \tau L)\frac{\left|f(x_n) - f(x_*)\right|^2}{\left|x_0 - x_*\right|^2}$$ Hint: use question 2.c)

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Justify that the map $F : \mathbf{R}_+^* \rightarrow S_n(\mathbf{R})$ is differentiable and that $$F'(1) = 2n(p(S))^\top p(S) - 2S^\top (p'(S))^\top p(S) - 2(p(S))^\top p'(S) S.$$

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Deduce, using the results of question 4, that $$\frac{n}{2(r-1)} F(r) \underset{r \rightarrow 1}{=} J(h) + o(1)$$

Let $f : (0,2) \cup (4,6) \rightarrow \mathbb{R}$ be a differentiable function. Suppose also that $f''(x) = 1$ for all $x \in (0,2) \cup (4,6)$. Which of the following is ALWAYS true?
(A) $f$ is increasing
(B) $f$ is one-to-one
(C) $f(x) = x$ for all $x \in (0,2) \cup (4,6)$
(D) $f(5.5) - f(4.5) = f(1.5) - f(0.5)$

Let $f : ( 0, 2 ) \cup ( 4, 6 ) \rightarrow \mathbb { R }$ be a differentiable function. Suppose also that $f ^ { \prime \prime } ( x ) = 1$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$. Which of the following is ALWAYS true?
(A) $f$ is increasing
(B) $f$ is one-to-one
(C) $f ( x ) = x$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$
(D) $f ( 5.5 ) - f ( 4.5 ) = f ( 1.5 ) - f ( 0.5 )$

Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$, defined as follows: $$f(x) = \begin{cases} (x-1)\min\left\{x, x^2\right\} & \text{if } x \geq 0 \\ x\min\left\{x, \frac{1}{x}\right\} & \text{if } x < 0 \end{cases}$$ Then, $f$ is
(A) differentiable everywhere.
(B) not differentiable at exactly one point.
(C) not differentiable at exactly two points.
(D) not differentiable at exactly three points.

Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as $$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$ Then which of the following statements is correct?
(A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$.
(B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$.
(C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.
(D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.

Let $f , g$ be differentiable functions on the real line $\mathbb { R }$ with $f ( 0 ) > g ( 0 )$. Assume that the set $M = \{ t \in \mathbb { R } \mid f ( t ) = g ( t ) \}$ is non-empty and that $f ^ { \prime } ( t ) \geq g ^ { \prime } ( t )$ for all $t \in M$. Then which of the following is necessarily true?
(A) If $t \in M$, then $t < 0$.
(B) For any $t \in M , f ^ { \prime } ( t ) > g ^ { \prime } ( t )$.
(C) For any $t \notin M , f ( t ) > g ( t )$.
(D) None of the above.

Let $f ( x ) , g ( x )$ be functions on the real line $\mathbb { R }$ such that both $f ( x ) + g ( x )$ and $f ( x ) g ( x )$ are differentiable. Which of the following is FALSE ?
(A) $f ( x ) ^ { 2 } + g ( x ) ^ { 2 }$ is necessarily differentiable.
(B) $f ( x )$ is differentiable if and only if $g ( x )$ is differentiable.
(C) $f ( x )$ and $g ( x )$ are necessarily continuous.
(D) If $f ( x ) > g ( x )$ for all $x \in \mathbb { R }$, then $f ( x )$ is differentiable.

Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable function such that $\frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } }$ is positive for all $x \in \mathbb { R }$, and suppose $f ( 0 ) = 1 , f ( 1 ) = 4$. Which of the following is not a possible value of $f ( 2 )$ ?
(A) 7 .
(B) 8 .
(C) 9 .
(D) 10 .

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be any twice differentiable function such that its second derivative is continuous and $$\frac { d f ( x ) } { d x } \neq 0 \text { for all } x \neq 0$$ If $$\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = \pi$$ then
(A) for all $x \neq 0 , \quad f ( x ) > f ( 0 )$.
(B) for all $x \neq 0 , \quad f ( x ) < f ( 0 )$.
(C) for all $x , \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } > 0$.
(D) for all $x , \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } < 0$.

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) + ( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(B) $( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) - ( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(C) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = ( 2 - a ) ^ { 2 }$
(D) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = - ( 2 + a ) ^ { 2 }$

Let $f ( x ) = x \sin \pi x , x > 0$. Then for all natural numbers $n , f ^ { \prime } ( x )$ vanishes at
(A) a unique point in the interval $\left( n , n + \frac { 1 } { 2 } \right)$
(B) a unique point in the interval $\left( n + \frac { 1 } { 2 } , n + 1 \right)$
(C) a unique point in the interval $( n , n + 1 )$
(D) two points in the interval $( n , n + 1 )$

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
If the function $\mathrm { e } ^ { - x } f ( x )$ assumes its minimum in the interval $[ 0,1 ]$ at $x = \frac { 1 } { 4 }$, which of the following is true?
(A) $f ^ { \prime } ( x ) < f ( x ) , \frac { 1 } { 4 } < x < \frac { 3 } { 4 }$
(B) $f ^ { \prime } ( x ) > f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(C) $f ^ { \prime } ( x ) < f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(D) $f ^ { \prime } ( x ) < f ( x ) , \frac { 3 } { 4 } < x < 1$

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V \mathrm {~mm} ^ { 3 }$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $\frac { V } { 250 \pi }$ is

Consider all rectangles lying in the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq x \leq \frac { \pi } { 2 } \text { and } 0 \leq y \leq 2 \sin ( 2 x ) \right\}$$
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
(A) $\frac { 3 \pi } { 2 }$
(B) $\pi$
(C) $\frac { \pi } { 2 \sqrt { 3 } }$
(D) $\frac { \pi \sqrt { 3 } } { 2 }$

Let $\mathbb { R }$ denote the set of all real numbers. Define the function $f : \mathbb { R } \rightarrow \mathbb { R }$ by
$$f ( x ) = \left\{ \begin{array} { c c } 2 - 2 x ^ { 2 } - x ^ { 2 } \sin \frac { 1 } { x } & \text { if } x \neq 0 \\ 2 & \text { if } x = 0 \end{array} \right.$$
Then which one of the following statements is TRUE?

(A)	The function $f$ is NOT differentiable at $x = 0$
(B)	There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval ( $0 , \delta$ )
(C)	For any positive real number $\delta$, the function $f$ is NOT an increasing function on the interval ( $- \delta , 0$ )
(D)	$x = 0$ is a point of local minima of $f$

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f ( x ) = \log _ { \mathrm { e } } x$ on the interval $[ 1,3 ]$ is
(1) $2 \log _ { 3 } e$
(2) $\frac { 1 } { 2 } \log _ { e } 3$
(3) $\log _ { 3 } e$
(4) $\log _ { e } 3$